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EURASIP Journal on Wireless Communications and NetworkingVolume 2008, Article ID 367287, 10 pages doi:10.1155/2008/367287 Research Article Differentially Encoded LDPC Codes—Part II: Gene

EURASIP Journal on Wireless Communications and Networking

Volume 2008, Article ID 367287, 10 pages

doi:10.1155/2008/367287

Research Article

Differentially Encoded LDPC Codes—Part II:

General Case and Code Optimization

Jing Li (Tiffany)

Electrical and Computer Engineering, Lehigh University, Bethlehem, PA 18015, USA

Correspondence should be addressed to Jing Li (Tiffany),jingli@ece.lehigh.edu

Received 19 November 2007; Accepted 6 March 2008

Recommended by Yonghui Li

This two-part series of papers studies the theory and practice of differentially encoded low-density parity-check (DE-LDPC) codes, especially in the context of noncoherent detection Part I showed that a special class of DE-LDPC codes, product accumulate codes, perform very well with both coherent and noncoherent detections The analysis here reveals that a conventional LDPC code, however, is not fitful for differential coding and does not, in general, deliver a desirable performance when detected noncoherently Through extrinsic information transfer (EXIT) analysis and a modified “convergence-constraint” density evolution (DE) method developed here, we provide a characterization of the type of LDPC degree profiles that work in harmony with differential detection (or a recursive inner code in general), and demonstrate how to optimize these LDPC codes The convergence-constraint method provides a useful extension to the conventional “threshold-constraint” method, and can match an outer LDPC code to any given inner code with the imperfectness of the inner decoder taken into consideration

Copyright © 2008 Jing Li (Tiffany) This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

With an increasingly mature status of the sparse-graph

coding technology in a theoretical context, the very pervasive

scope of their well-proven practical applications, and the

wide-scale availability of software radio, low-density

parity-check (LDPC) codes have become and continue to be a

favorable coding strategy for researchers and practitioners

Their superb performance on various channel models and

with various modulation schemes have been documented in

many papers While the existing literature has shed great light

on the theory and practice of LDPC codes, investigation was

largely carried out from a pure coding perspective, where

the prevailing assumption is that the synchronization and

channel estimation are handled perfectly by the front-end

receiver

In wireless communications, accurate phase estimation

may in many cases be very expensive or infeasible, which calls

for noncoherent detection Practical noncoherent detection

is generally performed in one of the two ways: inserting

pilot symbols directly in the coded and modulated sequence

to help track the channel (it is possible to insert either

pilot tones or pilot symbols, but the latter is found to be

more effective and is what of relevance to this paper), and employing differential coding Considering that the former may result in a nontrivial expansion of bandwidth especially

on fast-changing channels, many wireless systems adopt the latter, including satellite and radio-relay communications The problem we wish to investigate is: LDPC codes perform remarkably well with coherent detection, but how about their performance with noncoherent detection and noncoherent differential detection in particular? This series

of two-part papers aim to generate useful insight and engineering rules In Part I of the series [1], we considered

a special class of differentially encoded LDPC (DE-LDPC) codes, product accumulate (PA) codes [2] The outer code

of a (p(t + 2), pt) PA code is a simple, structured LDPC code

with left (variable) degree profileλ(x) =1/(t + 1) + t/(t + 2)x

and right (check) degree profileρ(x) = x t; and the inner code

is a differential encoder 1/(1 + D) We showed that, despite their simplicity, PA codes perform quite well with coherent detection as well as noncoherent differential detection [1] This motivates us, in Part II of this series of papers, to study the general case of differentially encoded LDPC codes The question of how LDPC codes perform with differential coding is a worthy one [3 6], and directly relates to other

interesting problems For example, what is the best strategy

to apply LDPC codes in noncoherent detection—should

differential coding be used or not? Modulation schemes such

as the minimum phase shift keying (MPSK) have equivalent

realizations in recursive and non-recursive forms; is one

form preferred over the other in the context of LDPC coding?

What other DE-LDPC configurations, besides PA codes, are

good for differential coding, and how to find them?

Since the conventional differential detector (CDD)

oating on two symbol intervals incurs a nontrivial

per-formance loss [7], and since multiple symbol differential

detectors (MSDD) [8] have a rather high complexity that

increases exponentially with the window size, we developed,

in Part I of this series of papers, a simple iterative differential

detection and decoding (IDDD) receiver, whose structure is

shown in [1, Figure 6] The IDDD receiver comprises a CDD

with 2-symbol observation window (the current and the

previous), a phase-tracking Wiener filter, a message-passing

decoder for the accumulator 1/(1 + D) [2], and a

message-passing decoder configured for the (outer) LDPC code The

CDD, coupled with the phase-tracking unit and the 1/(1 +

D) decoder, acts as the front-end, or, the inner decoder of

the serially concatenated system, and the succeeding LDPC

decoder acts as the outer decoder Soft reliability information

in the form of log-likelihood ratio (LLR) is exchanged

between the inner and the outer decoders to successively

refine the decision In the sequel, unless otherwise stated, we

take the IDDD receiver as the default noncoherent receiver in

our discussion of DE-LDPC codes

We study the convergence property of IDDD for a general

DE-LDPC code, through extrinsic information transfer

(EXIT) charts [9,10] A somewhat unexpected finding is

that, while a high-rate PA code yields desirable performance

with noncoherent (differential) detection, a general

DE-LDPC code does not We attribute the reason to the

mis-match of the convergence behavior between a conventional

LDPC code and a differential decoder This suggests that

conventional LDPC codes, while an excellent choice for

coherent detection, are not as desirable for noncoherent

detection It also gives rise to the question of what special

LDPC codes, possibly performing poorly in the conventional

scenario (such as the outer code of the PA code), may turn

out right for differential modulation and detection?

One remarkable property of LDPC codes is the

possibil-ity to design their degree profiles, through denspossibil-ity evolution

[11], to match to a specific channel or a specific inner code

[12–15] To make LDPC codes work in harmony with the

noncoherent differential decoder of interest, here we develop

a convergence-constraint density evolution method The

conventional threshold-constraint method [11,16] targets the

best asymptotic threshold, and the new method effectively

captures the interaction and convergence between the inner

and the outer EXIT curves through a set of “sample points.”

In that, it makes it possible to optimize LDPC codes to

match to an (arbitrary) inner code/modulation with the

imperfectness of the inner decoder/demodulator taken into

account Our study reveals that LDPC codes may be divided

in two groups Those having minimum left degree of≥2 are

generally suitable for a nonrecursive inner code/modulator

but not for a differential detector or any recursive inner code On the other hand, the LDPC codes that perform well with a recursive receiver always have degree-1 (and degree-2) variable nodes Further, when the code rate is high, these degree-1 and -2 nodes become dominant This also explains why high-rate PA codes, whose outer code has degree-1 and degree-2 nodes only, perform remarkably with (noncoherent) differential detection [1]

The channel model of interest here is flat Rayleigh fading channels with additive white Gaussian noise (AWGN), the same as discussed in Part I [1] Letr kbe the noisy signal at the receiver, let s k be the binary phase shift keying (BPSK) modulated signal at the transmitter, let n k be the i.i.d complex AWGN with zero mean and varianceσ2 = N0/2 in

each dimension, and letα k e jθ kbe the fading coefficient with Rayleigh distributed amplitudeα kand uniformly distributed phaseθ k We haver k = α k e jθ k s k+n k Throughout the paper,

θ k is assumed known perfectly to the receiver/decoder in the coherent detection case, and unknown (and needs to be worked around) in the noncoherent detection case Further, the receiver is said to have channel state information (CSI) if

α kknown (irrespective ofθ k), and no CSI otherwise

We consider correlated channel fading coefficients (so that noncoherent detection is possible) Applying Jakes’ isotropic scattering land mobile Rayleigh channel model, the autocorrelation ofα kis characterized by the 0th-order Bessel function of the first kind

Rk =1

and the power spectrum density (PSD) is given by

S( f ) = P π



1−f / f d

2, for| f | < f d, (2)

where f d T s is the normalized Doppler spread, f is the

frequency band,τ is the lag parameter, and P is a constant

that is dependent on the average received power given a specific antenna and the distribution of the angles of the incoming power

The rest of the paper is organized as follows.Section 2

evaluates the performance of a conventional LDPC code with noncoherent detection, and compare it with that of PA codes

Section 3 proposes the convergence-constraint method to optimize LDPC codes to match to a given inner code and particular a differential detector Section 4 concludes the paper

Part I showed that PA codes, a special class of DE-LDPC codes, perform quite well with coherent detection as well as noncoherent detection [1] This section reveals whether or not this also holds for general DE-LDPC codes, and the far subtly why

The analysis makes essential use of the EXIT charts [9,10], which are obtained through a repeated application

of density evolution at different decoding stages Although they were initially proposed solely as a visualization tool,

recent studies have revealed surprisingly elegant and useful

properties of EXIT charts Specifically, the convergence

prop-erty states that, in order for the iterative decoder to converge

successfully, the outer EXIT curve should stay strictly below

the inner EXIT curve, leaving an open tunnel between the

two curves The area property states that the area under

the EXIT curve, A = 10I e dI a, corresponds to the rate of

the code [10], whereI a andI e denote the a priori (input)

mutual information to and the extrinsic (output) mutual

information from a particularly subdecoder, respectively

When the auxiliary channel is an erasure channel and

the subdecoder is an optimal one, the relation is exact;

otherwise, it is a good approximation [10] The immediate

implication of these properties is that, to fully harness the

capacity (achievable rate) provided by the (noncoherent)

inner differential decoder, the outer code must have an EXIT

curve closely matched in shape and in position to that of the

inner code

With this in mind, we evaluate a few examples of

(DE-)LDPC codes (The computation of EXIT charts specific

to DE-LDPC codes with IDDD receiver is discussed in [1].)

We consider two configurations of the inner code:

(1) a differential decoder for 1/(1 + D); and

(2) a direct detector, that is, a BPSK detector;

and three configurations of the outer code:

(1) the outer code of a PA code, which has degree profile:

λ(x) =1

7+

6

7x, ρ(x) = x7;

(3)

(2) a (3,12)-regular LDPC code; and

(3) an optimized irregular LDPC code reported in [17],

whose threshold is 0.6726—about 0.0576 dB away

from the AWGN capacity—and whose degree profile is

γ(x) =0.1510x+0.1978x2+0.2201x6+0.03537+0.3958x29,

ρ(x) = x20.

(4) All three outer codes have rate 3/4, and the channel

is a correlated Rayleigh fading channel with AWGN and

normalized Doppler rate of f d T s =0.01.

The EXIT curves, plotted inFigure 1, demonstrate that

the outer code of the PA code and the differential decoder

match quite well, but a conventional LDPC code, regular or

irregular, will either intersect with the differential decoder,

causing decoder failure, or leave a huge area between

the curves, causing a capacity loss On the other hand,

LDPC codes, especially the (optimized) irregular ones,

agree very well with the direct detector This suggests that

(conventional) LDPC codes perform better as a single code

than being concatenated with a recursive inner code Put it

another way, an LDPC code that is optimal in the usual sense,

for example, BPSK modulation and memoryless channels,

may become quite suboptimal when operated together with

a recursive inner code or a recursive modulation, such

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

I e, /I a

I a,i /I e,o

0.35

0.29

0.24

0.19

0.15

0.12

0.085

0.055

0.026

0.013

0.0025

0.005

0.0076

f d T s =0.01, E b /N0=5.32, R =3/4

Differential code 1/(1 + D)

Fading channel Outer code of PA code

LDPC code (regular) LDPC code (irregular)

Regular LDPC Irregular LDPC

PA (outer)

Rayleigh channel

Di fferential code

Figure 1: EXIT curves of LDPC codes, the outer code of PA codes, differential decoder and the direct detector of Rayleigh channels Normalized Doppler rate 0.01, E b /N0 = 5.32 dB, code rate 3/4,

(3, 12)-regular LDPC code, and optimized irregular LDPC code withρ(x) = x20andγ(x) =0.1510x+0.1978x2+0.2201x6+0.03537+

0.3958x29

as a differential encoder On the other hand, not using differential coding generally requires more pilot symbols in order to track the channel well, especially on fast-fading environments Hence, it is fair to say that (conventional) LDPC codes that boast outstanding performance under coherent detection may not be nearly as advantageous under noncoherent detection, since they either suffer from performance loss (with differential encoding) or incur a large bandwidth expansion (without differential encoding)

In comparison, PA codes can make use of the (intrinsic)

differential code for noncoherent detection, and therefore present a better choice for bandwidth-limited wireless appli-cations

Before providing simulations to confirm our findings, we note that the EXIT curves of both inner codes inFigure 1are computed using perfect knowledge of the fading coefficients

We used this genie-aided case in the discussion, to rid off the artifact of coarse channel estimation and better contrast the differences between the recursive differential detector and the nonrecursive direct detector If the amplitude and phase information is to be estimated and handled by the inner code as in actual noncoherent detection, then the EXIT curve of the direct detector will show a small rising slope at the left end instead of being a flat straight line all the way through, and the EXIT curve of the differential decoder will also exhibit a deeper slope at the left end

Figure 2plots the BER performance curves of the same

three codes specified inFigure 1on Rayleigh channels with

noncoherent detection All the codes have data block size

K = 1 K and code rate 3/4 Soft feedback is used in

IDDD, the normalized Doppler spread is 0.01, and 2% or

4% pilot symbols are inserted to help track the channel

The two LDPC codes are evaluated either with or without

a differential inner code From the most power efficient

to the least power efficient, the curves shown are (i) PA

code with 4% of pilot symbols, (ii) PA code with 2% of

pilot symbols, (iii) BPSK-coded irregular LDPC code with

4% of pilot symbols, (iv) BPSK-coded regular LDPC code

with 4% of pilot symbols, (v) BPSK-coded irregular LDPC

code with 2% of pilot symbols, (vi) differentially encoded

irregular LDPC code with 4% of pilot symbols It is evident

that (conventional) LDPC codes suffer from a differential

inner code For example, with 4% of bandwidth expansion,

BPSK-coded irregular and regular LDPC codes perform

about 0.5 and 1 dB worse than PA codes at BER of 10−4,

respectively, but the differentially encoded irregular LDPC

code falls short by more than 2.2 dB Further, while the

irregular LDPC code (not differentially coded) is moderately

(0.5 dB) behind the PA code with 4% of pilot symbols, the

gap becomes much more significant when pilot symbols are

reduced in half For PA codes, 2% of pilot symbols remain

adequate to support a desirable performance, but they

become insufficient to track the channel for nondifferentially

encoded LDPC codes, causing a considerable performance

loss and an error floor as high as BER of 10−3 Thus, the

advantages of PA codes over (conventional) LDPC codes

are rather apparent, especially in cases when noncoherent

detection is required and when only limited bandwidth

expansion is allowed

PROPERTY

EXIT analysis and computer simulations in the previous

section show that a conventional LDPC code does not fit

differential coding, but special cases such as the the outer

code of PA codes do This raises more interesting questions:

what other (special) LDPC codes are also in harmony with

differential encoding? What degree profiles do they have? Is

it possible to characterize and optimize the degree profiles,

and how?

The fundamental tool to solve these questions lies in

convex optimization In [11], the optimization problem

of the irregular LDPC degree profiles on AWGN channels

was formulated as a duality-based convex optimization

problem, and an iterative method termed density evolution

was proposed to solve the problem In [16], a Gaussian

approximation was applied to the density evolution method,

which reduces the problem to be a linear optimization

problem Density evolution has since been exploited, in

different flavors and possibly combined with differential

evolution [18], to design good LDPC ensembles for a

vari-ety of communication channels and modulation schemes,

10−1

10−2

10−3

10−4

10−5

E b /N0 (dB)

K =1 K,R =3/4, f d T s =0.01, 10 iter

PA, 2%

PA, 4%

Irregular LDPC, 4%

Regular LDPC, 4%

Irregular LDPC, 2% Irregular LDPC, 4%, dif dec.

Figure 2: Comparison of PA codes and LDPC codes on fast-fading Rayleigh channels with noncoherent detection and decoding Solid line: PA codes, dashed lines: LDPC codes Code rate 0.75, data block size 1 K, filter length 65, normalized Doppler spread 0.01, 10 global iterations, and 4 (local) iterations within LDPC codes or the outer code of PA codes inside each global iteration

see, for example [12–15] and the references therein The results reported in these previous papers are excellent, but they almost exclusively aimed at the asymptotic threshold, namely, their cost functions were set to minimize the SNR threshold for a target code rate, or, equivalently, to maximize the code rate for a target SNR threshold This is well justified, since in these papers, the primary involvement of the channel

is to provide the initial LLR information to trigger the start

of the density evolution process

However, the problem we consider here is somewhat

different Our goal is to design codes that can fully achieve the capacity provided by the given inner receiver, and the noncoherent differential decoder in particular Considering that the inner receiver, due to the lack of channel knowledge

or other practical constraints, may not be an optimal receiver,

it is of paramount importance to control the interaction between the inner and the outer code, or, the convergence behavior as reflected in the matching of shape and position of the corresponding EXIT curves To emphasize the difference,

we thereafter refer to the conventional density evolution method as the “threshold-constraint” method, and propose

a “convergence-constraint” method as a useful extension to the conventional method

The key idea of the proposed method is to sample the inner EXIT curve and design an (outer) EXIT curve that matches with these sample points, or, “control points.” Suppose we choose a set ofM control points in the EXIT

plane, denoted as (v1,w1), (v2,w2), , (v M,w M) Let To(·)

be the input-output mutual information transfer function

of the outer LDPC code (whose exact expression of To

will be defined later in (17)), the optimization problem is

formulated as

max

Dv

i =1λ i =1, Dc

j =2ρ j =1



R =1−

D c

j =2ρ j / j

D v

i =1λ i /i | To



w k



≥ v k,k =1, 2, , M

 ,

(5)

whereR denotes the code rate of the outer LDPC code, and

λ iandρ idenote the fraction of edges that connect to variable

nodes and check nodes of degreei, respectively.

The formulation in (5) assumes that the LLR

mes-sages at the input of the inner and the outer decoder

are Gaussian distributed, and that the output extrinsic

mutual information (MI) of an irregular LDPC code

corresponds to a linear combination of the extrinsic MI

from a set of regular codes As reported in literature,

the Gaussian assumption for LLR messages is less not

far from reality on AWGN channels but less accurate

on Rayleigh fading channels [12] Nevertheless, Gaussian

assumption is used for several reasons The first reason

is simplicity and tractability Tracking and optimizing the

exact message pdf ’s involves tedious computation, which

is exacerbated by the fact the proposed new method is

governed by a set of control points, rather than a single

control point as in the conventional method Second, recall

that to compute EXIT curves inevitably uses the Gaussian

approximation Thus, it seems well acceptable to adopt

the same approximation when shaping and positioning an

EXIT curve Finally, characterizing and representing EXIT

curves using mutual information help stabilize the process

and alleviate the inaccuracy caused by Gaussian

approxi-mation and other factors As confirmed by many previous

papers as well as this one, the optimization generates very

good results in spite of the use of the Gaussian

approxima-tion

Below we detail the convergence-constraint design method

formulated in (5) We conform to the notations and the

graphic framework presented in [16] Letλ(x) =D v

i =1λ i x i −1

andρ(x) =D c

i =2ρ i x i −1be the degree profiles from the edge

perspective, where D v and D c are the maximum variable

node and check node degrees, andλ iandρ iare the fraction

of edges incident to variable nodes and check nodes of degree

i Similarly, let λ (x) =D v

i =1λ  i x i −1andρ (x) =D c

i =2ρ  i x i −1be the degree profiles from the node perspective LetR be the

code rate The following relation holds:

λ  i =D λ v i /i

j =1λ j / j, ρ



i =D ρ c i /i

j =2ρ j / j, R =1−

D v

i =1λ i /i

D c

j =1ρ j / j .

(6) Let superscript (l) denote the lth LDPC decoding iteration,

and subscript v and c denote the quantities pertaining to

variable nodes and check nodes, respectively Further, define

two functions that will be useful in the discussion

I(x) =1−

∞

−∞

1

√

2πx e

1 +e − z

dz, (7)

φ(x) =

⎧

⎪

⎪

1− √1

4πx

tanhz

2e −(− x)2/4x dz, x > 0,

(8)

Function I(x) maps the message mean x to the

corre-sponding mutual information (under Gaussian assumption), andφ(x) helps describe how the message mean evolves in

tanh(y/2) operation, where y follows a Gaussian distribution

with meanx and variance 2x.

The complete design process takes a dual constraint optimization process that progressively optimizes variable node degree profileλ(x) and check node degree profile ρ(x)

based on each other Despite the duality in the formulation and the steps, optimizing λ(x) is far more critical to the

code performance than optimizingρ(x), largely because the

optimal check node degree profile are shown to follow the concentration rule [16]:

ρ(x) = Δx k+ (1− Δ)x k+1 (9)

It is therefore a common practice to presetρ(x) according to

(9) and code rateR, and optimize λ(x) only For this reason,

below we focus our discussion on optimizingλ(x) for a given ρ(x) Interested readers can formulate the optimization of ρ(x) in a similar way.

3.2.1 Threshold-constraint method (optimizing λ(x))

Under the assumption that the messages passed along all the edges are i.i.d and Gaussian distributed, the average messages variable nodes receive from their neighboring check nodes follow a mixed Gaussian distribution From (l −1)th iteration tolth local iteration (in the LDPC decoder),

the mean of the messages associated with the variable node,

m v, evolves as

m(l)

v =

D v



i =2

λ iNm(v,i l), 2m(v,i l)



(10)

=

D v



i =2

λ i φ



m0+ (i −1)

D c



j =2

ρ j φ −1

1−1− m(v l −1)

j −1 , (11) wherem0 denotes the mean of the initial messages received from the inner code (or the channel) Let us define

h i(m0,r) =Δφ



m0+ (i −1)

D c



j =2ρ j φ −1

1−(1− r) j −1

,

h

m0,r Δ

= Dv

i =2λ i h i



m0,r

.

(12) Then (11) can be rewritten as

r l = h

m0,r l −1



=

D v



=

λ i h i



m0,r l −1



The conventional threshold-constraint density evolution

guarantees that the degree profile converges asymptotically

to the zero-error state at the given initial message meanm0

This is achieved by enforcing [16]

r > h

m0,r , ∀ r ∈0,φ

m0



Viewed from the EXIT chart, the threshold-constraint

method has implicitly used a control point (v, w) = (1,

I(m0)), such that the resultant EXIT curve will stay below

it

3.2.2 Convergence-constraint method (optimizing λ(x))

The proposed convergence-constraint method extends the

conventional threshold-constraint method by introducing

a set of control points, which may be placed in arbitrary

positions in the EXIT plane, to control the shape and the

position of the EXIT cure Each control point (v, w) ∈

[0, 1]2ensures that the EXIT curve will, at the input a priori

mutual informationw, produce extrinsic mutual

informa-tion greater than v This is reflected in the optimization

process by changing (14) to

r ∗ > h

m0,r ∗ , ∀ r ∗ ∈0,φ

m0



wherer ∗(≥0) is the threshold value that satisfiesT0(w) ≥ v.

We can reformulate the problem as follows: for a given check

node degree profile ρ(x) and a control point (v, w) in the

EXIT chart, where 0≤ v, w, ≤1,

max

Dv

i =1λ i =1

D v



i =1

λ i

i,

subject to: (i)

D v



i =1

λ i =1,

(ii)

D v



i =1

λ i



h i



m0,r

− r

< 0, ∀ r ∈r ∗,φ

m0



, (16) wherem0=I−1(w) and r ∗satisfies

To(w) =Δ

D v



i =1

λ  iI



i

D c



j =2

ρ j φ −1

1−1− r ∗j −1

Apparently, whenv =1, we getr ∗ =0, and the case reduces

to that of the conventional threshold-constraint design

Hence, given a set of M control points, (v1,w1),

(v2,w2), , (v M,w M), where 0 ≤ v1 < v2 < · · · < v M ≤1

and 0 ≤ w1 ≤ w2 ≤ · · · ≤ w M ≤ 1, one can combine

the constraints associated with each individual control point

and perform joint optimization, to control the shape and the

position of the resulting EXIT curve Specifically, when the

set of control points are proper samples from the inner EXIT

curve, the resultant EXIT curve represents an optimized

LDPC ensemble that matches to the inner code

3.2.3 Linear programming

The basic idea of convergence-constraint design, as discussed before, is simple Complication arises from the fact that constraint (ii) in (16) is a nonlinear function ofλ i’s Further-more, observe that the determination of the optimization range, or, the computation of r ∗ from (17), requires the knowledge ofλ(x), which is yet to be optimized One possible

approach to overcome this chicken-and-egg dilemma is

to attempt an approximated λ(x) in (17) to compute r ∗ Specifically, we propose accounting for the two lowest degree variable nodes λ i1 and λ i2, and approximating the degree profile as



λ(x) = λ i1x i1−1+λ i2x i2−1+O

λ i2 +1x i2

≈ λ i1x i1−1+

1− λ i1



x i1 (18)

in (17) First, this approximatedλ(x) is only used in (17) to tentatively determine r ∗, so that the optimization process can get started The exact λ(x) in (16), (i) and (ii), is to

be optimized Second, the value ofi1andλ i1(orλ  i1) in the approximatedλ(x) is calculated in one of the following two

ways

Case 1 A conventional LDPC ensemble has i1 = 2, that is,

no degree-1 variable nodes This is because the outbound messages from degree-1 variable nodes do not improve over the message-passing process In that case, we consider only degree-2 and 3 nodes (λ i1= 2 and λ i2= 3), upper bound the percentage of degree-2 nodes withλ ∗2, and treat all the rest

as degree-3 nodes The stability condition [11, 16] states that there exists a value ξ > 0 such that, given an initial

symmetric message densityP0 satisfying0

−∞ P0(x)dx < ξ,

then the necessary and sufficient condition for the density evolution to converge to the zero-error state isλ (0)ρ (1)<

e γ, whereγ = −Δ log(∞

−∞ P0(x)e − x/2 dx) Applying the stability

condition on Gaussian messages with initial mean value

m0, we getγ = m0/4 and λ ∗2 = e m0/4 /D c

j =2(j −1)ρ j, or equivalently,

λ ∗2(w) = eI

−1 (w)/4

D c

j =2ρ j(j −1). (19)

It should be noted that not all values of w k from the M preselected control points are suitable for (19) in computing λ ∗2 Since the stability condition ensures the

asymptotic convergence to the zero-error state for a given

input messages, λ2 ≤ λ ∗2(w ∗) is valid and required only when the output mutual information will approach 1 at the input mutual informationw ∗ What this implies in sampling the inner EXIT curve is that, at least one control point, say, the rightmost point (v M,w M), should roughly satisfy the requirement: (v M,w M)≈ (1,w M) This value ofw M is then used in (19) to computeλ ∗2 = λ ∗2(w M), which is subsequently used inλ(x) ≈ λ ∗

2x + (1 − λ ∗2)x2to computer ∗from (17).r ∗

will then be applied to all the control points from 1 toM.

Checks Bits Checks Bits

Error

Error

LDPC Di fferential encoder

p

q

Figure 3: Defect forλ 1> 1 − R When the four bits associated with

the solid circles flip altogether, another valid codeword results, and

the decoder is unable to tell (undetectable error)

It is also worth mentioning that when a Gaussian

approximation is used on the message pdf ’s, the stability

condition reduces to

λ ∗2(w) = eI

−1 (w)/4

D c

j =2(j −1)ρ j, (20) which is a weaker condition than (19) Since we use Gaussian

approximation primarily for the purpose of complexity

reduction, unnecessary application is therefore avoided

Thus (19) rather than (20) is used in our design process

Case 2 Consider the case when an LDPC code is iteratively

decoded together with a differential encoder, or, other

recursive inner code or modulation with memory Since

the inner code imposes another level of checks on all the

variable nodes, degree-1 variable nodes in the outer LDPC

code will get extrinsic information from the inner code, and

their estimates will improve with decoding iterations Thus,

without loss of generality, we let the first and the second

nonzero λ i’s be λ1 and λ2 No analytical bounds onλ1 or

λ2 were reported in literature for this case We propose to

boundλ 1byλ 1 ≤1− R, where R is the code rate (the exact

code rate is dependent on the optimization result, and may

be slightly different from the target code rate) The rational

is that, if λ 1 > 1 − R, then there exist at least two

degree-1 variable nodes, say the pth node and the qth node, which

connect to the same check When the LDPC code operates

alone, these two variable nodes are apparently useless and

wasteful, and can be removed altogether When the LDPC

code is combined with an inner recursive code, as shown

in Figure 3, these two degree-1 variable nodes will cause a

minimum distance of 4 for the entire codeword, irrespective

of the code length Using this empirical bound onλ1, we can

employ the approximationλ(x) =(1− R)+Rx in (17), which

leads to the computation of (a lower bound for)r ∗ Code

optimization as formulated by the convergence-constraint

method can thus be solved using linear programming

It is rather expected that the choice of the control points

directly affects the optimization results The set of control

points need not be large—in fact, an excessive number of control points actually makes the optimization process con-verge slow and at times concon-verge poor We suggest choosing

3 to 5 control points that can reasonably characterize the shape of the inner EXIT curve Our experiments show that the proposed method generates EXIT curves with a shape matching very well to what we desire, but the position

is slightly lower, indicating that the resultant code rate is slightly pessimistic This can be compensated by presetting the control points slightly higher than we actually want them

to be

For complexity concerns, instead of performing dual opti-mization, we apply the concentration theorem in (9) and preselectρ(x) that will make the the average column weight

to be approximately 3 The left degree profileλ(x) is

opti-mized through the convergence-constraint method discussed

in the previous subsection We now discuss some observa-tions and findings from our optimization experiments First, the LDPC ensemble optimal for differential coding always contains degree-1 and degree-2 variable nodes For high rate codes above 0.75, these nodes are dominant, and

in some cases, are the the only types of variable nodes in the degree profile For medium rates around 0.5, there exist also

a good portion of high-degree variable nodes Considering that the outer code of a PA code has only degree-1 and degree-2 variable nodes,λ(x) =(1− R)/(1 + R) + (2R/(1 + R))x, where R ≥ 1/2 is the code rate, it is fair to say

that PA are (near-)optimal at high rates, but less optimal

at medium rates (the optimized LDPC ensemble contains slightly different degree distribution than that of the PA code, the difference is very small in either asymptotic thresholds or finite length simulations) This is actually well reflected in the EXIT charts As rate 3/4 (seeFigure 1), the area between the outer code of the PA code and the inner differential code is very small, leaving not much room for improvement

In comparison, at rate around 0.5 (seeFigure 4), the area becomes much bigger, indicating that an optimized outer code could acquire more information rate for the same SNR threshold, or, for the same information rate, achieve a better SNR threshold

The optimization result of a target rate 0.5 is shown in

Figure 4 We consider an inner differential code, operating

at 0.25 dB on a f d T s = 0.01 Rayleigh fading channel, and

decoded using the noncoherent IDDD receiver with the help

of using 10% pilot symbols The optimzed LDPC ensemble has code rateR =0.5037 and degree profile

λ(x) =0.0672 + 0.4599x + 0.0264x8+ 0.0495x9

+ 0.0720x10+ 0.0828x11+ 0.0855x12

+ 0.0807x13+ 0.0760x14,

ρ(x) = x5.

(21)

We see that the two EXIT curves match very well with each other Here the inner EXIT curve is computed through Monte Carlo simulations, when the sequences are taken in

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

I e,

/I a

I a,i /I e,o

Code design

f d T s =0.01

1.26 dB, 10% pilots

0.25 dB, 10% pilots

R =0.5037, optimized LDPC

R =0.5, outer code of PA codes

Figure 4: EXIT chart of a rate 0.5 LDPC ensemble optimized

using convergence-evolution for differential coding Normalized

Doppler rate is 0.01, 10% of pilot symbols are assumed to assist

noncoherent differential detection Degree profile of the optimized

LDPC ensemble:λ(x) =0.0672 + 0.4599x + 0.0264x8+ 0.0495x9+

0.0720x10+ 0.0828x11+ 0.0855x12+ 0.0807x13+ 0.0760x14,ρ(x) = x5

blocks ofN =106bits, and the power penalty due to the pilot

symbols is also compensated for

The optimized LDPC ensemble requires 0.25 −

10 log10(0.5037) =3.2283 dB asymptotically, in order for the

iterative process to converge successfully Compared to a rate

0.50 PA code which requires 1.26 −10 log10(0.5) =4.2703 dB

(Figure 4), the optimized LDPC ensemble is about 1.04 dB

better asymptotically However, as the tunnel between the

inner and the outer EXIT curves becomes more narrow, the

message-passing decoder takes a larger number of iterations

to arrive at the zero-error state The increased computing

complexity and processing time are the price we pay for

reaching out to the limit

The optimized LDPC ensemble is good in the asymptotic

sense, that is, with infinite or very long code lengths In

practice, we are also concerned with finite-length

imple-mentation or individual code realization According to the

concentration rule, at long lengths, all code realizations

perform close to each other, and they all tend to converge

to the asymptotic threshold as length increases with bound

At short lengths, however, the concentration rule fails and

the performance may vary rather noticeably from one code

realization to another Good realizations have improved

neighborhood condition than others, including a larger

girth (achieved, e.g., through the edge progressive growth

algorithm), a smaller number of short cycles, or a smaller

trapping set

Figure 5 simulates the optimized rate-0.5037 LDPC

code with differential encoding and noncoherent differential

10 0

10−1

10−2

10−3

10−4

E b /N0 (dB) Opt LDPC, dif dec., ideal, 10%, 15 iter Opt LDPC, dif dec., 10%, 5, 10, 15 iter

PA, 10%, 15 iter Conv LDPC, non-dif dec., 10%, 15 iter

Analytical threshold

(64 K, 32 K), optimized LDPC,f d T s =0.01

Figure 5: Simulations of optimized LDPC code with differential coding and iterative differential detection and decoding Code rate 0.5037, normalized Doppler rate 0.01, 10% pilot insertion, degree profileλ(x) =0.0672 + 0.4599x + 0.0264x8+ 0.0495x9+ 0.0720x10+

0.0828x11+ 0.0855x12+ 0.0807x13+ 0.0760x14, andρ(x) = x5, 15 (global) iterations each with 6 (local) iterations in the outer LDPC decoding

decoding The Rayleigh channel and the inner differential decoder (the IDDD receiver) are the same as discussed in

Figure 4 We chose a long codeword length of N = 64 K

to test how well the simulation agrees with the analytical threshold As mentioned before, a large number of iterations (e.g., 100 iterations) is preferred to fully harness the code gain, but considering the complexity and delay affordable in

a practical system, we simulated only 15 iterations In the figure, the leftmost curve corresponds to the optimized DE-LDPC code using ideal detection (perfect knowledge on the fading phases and amplitudes), but with 10% pilot symbols These wasteful pilot symbols are included in this coherent detection case to offset the curve, and to compare fairly with all the other noncoherently detected curves with 10% of pilot insertion The three circled curves to the right of this ideal detection curve correspond to the noncoherent performance using iterative differential detection and decoding at the 5th, 10th, and 15th iteration We see that the performance of the optimized differentially encoded LDPC code performs only about 0.3 dB worse than the coherent detection, which

is very encouraging Further, the simulated performance

is only 0.75 dB away from the asymptotic threshold of

3.23 dB (discussed before), showing a good theory-practice

agreement

For reference, we also plot inFigure 5the performance of

a PA code and a conventional LDPC code without differential coding (recall that conventional LDPC codes perform worse with differential coding than without), both having code rate around 0.5 and both noncoherently detected We see that the

PA code outperforms the conventional LDPC code by 1.5 dB,

but the optimized DE-LDPC code outperforms the PA code

by another 1.4 dB!

Part I of this two-part series of papers [1] studied product

accumulate codes, a special case of differentially encoded

LDPC codes, with coherent detection and especially

nonco-herent detection It showed that PA codes perform very well

in both cases Here in Part II, we generalize the study from

PA codes to an arbitrary differentially encoded LDPC code

The remarkable performance of LDPC codes with

coher-ent detection has been extensively studied, but much less

work has been carried out on noncoherently detected LDPC

codes In general, a noncoherently detected system may or

may not employ differential encoding The former leads to a

differential encoding and noncoherent differential detection

architecture, and the latter requires the insertion of (many)

pilot symbols in order to track the (fast-changing) channel

well A rather unexpected finding here is that a conventional

LDPC code actually suffers in either case: in the former

it was because of an EXIT mismatch between the (outer)

LDPC code and the (inner) differential code, and in the latter

it was because of the large bandwidth expansion Here by

conventional we mean the LDPC code that delivers a superb

performance in the usual setting with coherent detection and

possibly channel state information

Further investigation shows that it is not only possible,

but highly beneficial, to optimize an LDPC code to match to

a differential decoder The optimization is achieved through

a new convergence-constraint density evolution method

developed here The resultant optimized degree profiles are

rather nonconventional, as they contain (many) degree-1

and -2 variable nodes This is in sharp contrast to the

conventional LDPC case (i.e., coherent detection) where

degree-1 variable nodes are deemed highly undesirable

The effectiveness of the new DE method is confirmed

by the fact that the optimized DE-LDPC code brings an

additional 1.4 dB and 2.9 dB, respectively, over the existing

PA code and the conventional LDPC code (when

nonco-herent detection is used) The proposed DE optimization

procedure is very useful It provides a practical way to tune

the shape and the position of an EXIT curve, and can

therefore match an LDPC code to virtually any front-end

processor, with the imperfectness of the processor taken into

explicit consideration

We conclude by stating that LDPC codes can after all

perform very well with differential encoding (or any other

recursive inner code or modulation), but the degree profiles

need to be carefully (re)designed, using, for example, the

convergence-constraint density evolution developed here,

and one should expect the optimized degree profile to

contain many degree-1 (and degree-2) variable nodes

ACKNOWLEDGMENTS

This research work supported in part by the National

Science Foundation under Grant no CCF-0430634 and

CCF-0635199, and by the Commonwealth of Pennsylvania through the Pennsylvania Infrastructure Technology Alliance (PITA)

REFERENCES

[1] J Li, “Differentially-encoded LDPC codes: part I—special case

of product accumulate codes,” to appear in EURASIP Journal

on Wireless Communications and Networking

[2] J Li, K R Narayanan, and C N Georghiades, “Product accumulate codes: a class of codes with near-capacity

perfor-mance and low decoding complexity,” IEEE Transactions on Information Theory, vol 50, no 1, pp 31–46, 2004.

[3] V T Nam, P.-Y Kam, and Y Xin, “LDPC codes with BDPSK and differential detection over flat Rayleigh fading channels,”

in Proceedings of the 50th IEEE Global Telecommunications Conference (GLOBECOM ’07), pp 3245–3249, Washington,

DC, USA, November 2007

[4] H Tatsunami, K Ishibashi, and H Ochiai, “On the per-formance of LDPC codes with differential detection over

Rayleigh fading channels,” in Proceedings of the 63rd IEEE Vehicular Technology Conference (VTC ’06), vol 5, pp 2388–

2392, Melbourne, Victoria, Australia, May 2006

[5] M Franceschini, G Ferrari, R Raheli, and A Curtoni, “Serial concatenation of LDPC codes and differential modulations,”

IEEE Journal on Selected Areas in Communications, vol 23,

no 9, pp 1758–1768, 2005

[6] J Mitra and L Lampe, “Simple concatenated codes using differential PSK,” in Proceedings of the 49th IEEE Global

Telecommunications Conference (GLOBECOM ’06), pp 1–6,

San Francisco, Calif, USA, November 2006

[7] M C Valenti and B D Woerner, “Iterative channel estimation and decoding of pilot symbol assisted turbo codes over

flat-fading channels,” IEEE Journal on Selected Areas in Communi-cations, vol 19, no 9, pp 1697–1705, 2001.

[8] M Peleg and S Shamai, “Iterative decoding of coded and interleaved noncoherent multiple symbol detected DPSK,”

Electronics Letters, vol 33, no 12, pp 1018–1020, 1997.

[9] S ten Brink, “Convergence behavior of iteratively decoded

parallel concatenated codes,” IEEE Transactions on Communi-cations, vol 49, no 10, pp 1727–1737, 2001.

[10] A Ashikhmin, G Kramer, and S ten Brink, “Extrinsic information transfer functions: model and erasure channel

properties,” IEEE Transactions on Information Theory, vol 50,

no 11, pp 2657–2673, 2004

[11] T J Richardson, M A Shokrollahi, and R L Urbanke,

“Design of capacity-approaching irregular low-density

parity-check codes,” IEEE Transactions on Information Theory, vol 47,

no 2, pp 619–637, 2001

[12] J Hou, P H Siegel, and L B Milstein, “Performance analysis and code optimization of low density parity-check codes on

Rayleigh fading channels,” IEEE Journal on Selected Areas in Communications, vol 19, no 5, pp 924–934, 2001.

[13] A Shokrollahi and R Storn, “Design of efficient erasure codes with differential evolution,” in Proceedings of the IEEE

Interna-tional Symposium on Information Theory, p 5, Sorrento, Italy,

June 2000

[14] S ten Brink, G Kramer, and A Ashikhmin, “Design of low-density parity-check codes for modulation and detection,”

IEEE Transactions on Communications, vol 52, no 4, pp 670–

678, 2004

[15] R.-R Chen, R Koetter, U Madhow, and D Agrawal, “Joint noncoherent demodulation and decoding for the block fading

channel: a practical framework for approaching Shannon

capacity,” IEEE Transactions on Communications, vol 51,

no 10, pp 1676–1689, 2003

[16] S.-Y Chung, T J Richardson, and R L Urbanke, “Analysis

of sum-product decoding of low-density parity-check codes

using a Gaussian approximation,” IEEE Transactions on

Infor-mation Theory, vol 47, no 2, pp 657–670, 2001.

[17] http://lthcwww.epfl.ch/research/

[18] R Storn and K Price, “Differential evolution—a simple and

efficient heuristic for global optimization over continuous

spaces,” Journal of Global Optimization, vol 11, no 4, pp 341–

359, 1997